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Full text of "Treatise On Analysis Vol-Ii"

94       XII    TOPOLOGY AND TOPOLOGICAL ALGEBRA

for 0 ^ i <^ n. For each real-valued function/defined on I, the Lagrange interpolation
polynomial of/corresponding to (a^) is the polynomial

(a) The mapping /i->P«(/) is a continuous linear mapping of the Banach space
<^(I) into itself. The sequence of polynomials Pn(/) converges uniformly to / on I
for allfe ^(1) if and only if the sequence of norms ||PJ of these linear mappings is
bounded (use (12.16.5) and (7.5.5)). Show that

(b) Take am—iln for 0<^*<^/?. Show that the sequence (||PB||) tends to +00.
Deduce that, for this choice of the aln, there exists a function /0 e ^(1) such that the
sequence of polynomials Pn(/o) is not uniformly bounded on I (use Problem 14).

16.    Let/be a continuous real-valued function on an open interval I in R, having a right-
hand derivative /£ (8.4) at every point of I.

(a)    Show that the set of points x e I such that/5 is bounded in some neighborhood
of x is a dense open set in I (use (12.16.2)).

(b)    Show that the set of points x e I at which f& is continuous is the complement of
a meager set in I (cf. Problem 3). Deduce that/has a derivative at all points of the
complement of a meager set in I (cf. Section 8.6, Problem 2).

17.    (a)   In the Banach space #(I), where I = [0, 1], let An be the set of functions / such
that for some x e 0, 1 — l//i', depending on /, we have   \f(x') ~-/(A:)| :g n \x' — x\
for all x' e ]x, x H- l/fl[. Show that An is nowhere dense in ^(1). (Notice first that in
tf(I) every ball contains a function which has a bounded right-hand derivative at all
points of I. Secondly, observe that for each e > 0 and each integer m, there exists a
function g e ^(1) with a right-hand derivative at every point of [0,1 [ and such that
\g(x)\ ^ e and \g'd(x)\ £ m at all x e [0, 1[.)

(b) Deduce from (a) that the set A of functions /e <^(I) such that, for at least one
point x e [0,1[ (depending on/),/has a right-hand derivative at Jt, is meager in ^(1).
Consequently, the complement of this set, that is to say the set of all /e ^(1) having
no right-hand derivative at any point of [0, If, is dense in *^(I) (cf. Section
8.4, Problem 1).

18.    Let/be a real-valued function defined on an open set A <= R2, such that for each point
(*o, JVo) e A the function /(-, y0) is continuous at x0 and the function /Ot0, ') is
continuous and differentiate at the point y0. Show that there exists a meager set M
in A such that the derivative D2/ is continuous on A — M (use Problem 8(b) and
Problems).

19.    Let E be a complex Banach space and E' its dual, which is a Banach space with respect
to the norm (5.7.1). Show that a weakly analytic mapping z\~*fz of an open set
A c C into E' is in fact analytic in the sense of (9.3) (with respect to the norm under
consideration). (Use the formula (12.16.6.6) and the Banach-Steinhaus theorem.)
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